Math 521, Lecture 2 - Spring 2009
Advanced Calculus
Homework 1
Exercice 1
(1) Prove that there is no rational number q∈Qsuch that q2= 3.
(2) Prove that the subset A:= {x∈Q|x2<3} ⊂ Qis bounded above in Q; and prove that A contains
no largest number.
Exercice 2 Prove that the relation ≤on Rsatisfies the three following properties
(1) for any x∈R,x≤x,
(2) for any x, y ∈R, if x≤yand y≤xthen x=y,
(3) for any x, y, z ∈R, if x≤yand y≤zthen x≤z.
Exercice 3 Let Aand Btwo nonempty subsets of Rsuch that A⊂B. Show that if Bis bounded above
then sup Aexists and sup A≤sup B. Is something similar can be said for inf Aand inf B?
Exercice 4 Do the exercice 1 from chapter 1 of the textbook.
Exercice 5 Find the sup and the inf (in case it exists) of the following substets of R:A={(−1)n, n ∈N},
B= [0,1),C= [−1,2] ∩[0,1],D= [−1,2] ∩(2,3],E= [−1,0] ∪(4,5),F= [2,+∞)c,G={x;x=
(−1
2)m−3
n, m, n ∈N∗}.
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